Help me to understand the time component of mass

Inertial mass in classical mechanics can be measured by the relation between force and acceleration. So it is necessary some time to make the measurement, but it doesn't mean it is a function of time.

For the relativistic equation, $E = m$ for the frame where the mass is at rest. Velocity of light is only a constant.

First of all, it is generally a bad idea to divide vectors. Write the equation as $m vec{a} = vec{F}$ or $vec{a} = vec{F} / m$. The mass is a scalar number (it has no time component) and describes the amount of matter or inertia. It does not change in the dynamics.

The $c^2$ term is the conversion constant for going back and forth between mass and energy units. It has nothing at all to do with Newton's $vec F = m vec a$ law.

In $m = F/a$ as written, the mass term is not changing that is, varying the acceleration does not magically cause the mass to change.